Newton s Law of Univesal avitation and the ale iniple RODOLO A. RINO July 0 Eletonis Enginee Degee fo the National Univesity of Ma del lata - Agentina (odolfo_fino@yahoo.o.a) Ealie this yea I wote a pape entitled ale atos and the ale iniple. In that pape I foulated a new law whih desibes a nube of fundaental quantu ehanial laws and pat of Einstein s theoy of elativity. The pupose of this atile is to show that this theoy also pedits Newton s law of univesal gavitation. Thus this new foulation an be extended to lassial ehanis. Keywods: Newton s law of univesal gavitation, Coulob s law, lank hage, lank length, lank tie, lank ass, lank aeleation, lank foe.. Intodution In a pevious atile [] I intodued the sale piniple o sale law though the following atheatial elationship () ale piniple o sale law n [ ] Whee a),, and ae physial quantities of idential diension (suh as Length, Tie, Mass, Tepeatue, et), o b) and ae physial quantities of diension o diensionless onstants while and ae physial quantities of diension o diensionless onstants. Howeve, if and ae diensionless onstants then and ust have diensions and vievesa. The physial quantities an be vaiables, onstants, diensionless onstants, diffeentials, deivatives (inluding Laplaians), integals, vetos, any atheatial opeation between the pevious quantities, et. Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino.
(e.g.: and ould be quantities of Mass while and ould be quantities of Length). ) The elationship is one of thee possibilities: less than o equal to inequation ( ), o equal to - equation (), o a geate than o equal to inequation ( ). d) is a diensionless sale fato. This fato ould be a eal nube, a oplex nube, a eal funtion o a oplex funtion (stitly speaking eal nubes ae a patiula ase of oplex nubes). The sale fato ould have oe than one value fo the sae elationship. In othe wods a sale fato an be a quantu nube. e) n and ae integes 0,,,, (In geneal these two nubes ae diffeent. e.g. : n and. e.g. : n and. n and annot be both zeo in the sae elationship). It is wothy to eak that so fa these integes ae not geate than, howeve this ould hange in the futue.. The lank oe In this setion I shall deive the lank foe fo two known laws. In subsetion. I stat the analysis fo the Coulob s law while in subsetion. the stating point is Newton s seond law of otion. The esults of these two independent deivations ae idential as it should be. The lank foe is used in setion to deive the law of univesal gavitation.. Deivation of the lank oe based on Coulob s Law We shall deive the lank foe fo Coulob s law E qq (Coulob s law) () π ε 0 We shall substitute both hages q and q with the lank hage and the distane with the lank length L, this yields L () Whee ε 0 h (lank hage) () Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino.
L h (lank length) (5) π π ε 0 ε 0 h h π (6) inally the lank foe is (lank oe) (7). Deivation of The lank oe based on Newton s eond Law of Motion We shall deive the lank foe fo Newton s seond law of otion a (8) The lank foe an also be defined as M a (9) Whee lank foe M lank ass a lank aeleation The lank ass is defined as M h π (0) The lank aeleation an be defined as a T () Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino.
Whee speed of light in vauu T lank tie T h π 5 () ubstituting T in equation () by the seond side of equation () yields 5 π a h π h 7 a () () ubstituting M and espetively yields a in equation (9) with the seond side of equations (0) and () h π π h 7 (5) inally the expession fo the lank foe is (lank oe) (6). Deivation of the Univesal Law of avitation In this setion I shall deive Newton s law of univesal gavitation fo the sale piniple. In 687 Isaa Newton published his inipia whee he stated his univesal law of gavitation as follows (Newton s law of univesal gavitation) (7) Whee avitational foe between two any bodies of asses and (this foe is also known as foe of univesal gavitation, gavity, gavity foe, foe of gavitational attation, foe of gavity, Newtonian foe of gavity, foe of univesal utual gavitation, et.) avitational onstant (also known as onstant of gavitation, onstant of gavity, Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino.
gavitational foe onstant, univesal onstant of gavity, univesal gavitational onstant, Newtonian gavitational onstant, et.) ass of body ass of body distane between the entes of body and body We stat the analysis by obseving that the nueato of equation (6) [o equation (7)] is. This suggests that the equation we ae looking fo should oiginate fo the podut of two elativisti enegies suh as E and E. We also notie that the fato of the podut should anel out with the nueato of equation (6). Thus, taking into onsideation these fats, we daw the following sale table Wok Wok Enegy (elativisti) Enegy (elativisti) W W E E TABLE : This sale table is used to deive Newton s law of univesal gavitation. In suay, the quantities shown in Table ust be defined as follows W (Wok done by ) (8) W (Wok done by ) (9) E (Relativisti enegy of body ) (0) E (Relativisti enegy of body ) () o the table we establish the following elationship W W E E () Replaing the vaiables W, W, E and E by equations (8), (9), (0) and () espetively we get () () Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino. 5
(5) If we obtain the Newton s law of univesal gavitation (see equation 7). Thus we have poved that Newton s law of univesal gavitation an be deived fo the sale piniple. It is wothy to eak that the gavitational onstant is not the sale fato as one ight be tepted to think. ale fatos ae diensionless while the gavitational onstant is not. Now let us expess equation () in the fo of the sale piniple (6) Copaing equation (6) with equation () we find that equation (6) has the following fo (7) Whee n Thus we have poved that Newton s law of univesal gavitation obeys the sale piniple.. Conlusions In this pape we have foulated a elativisti quantu ehanial deivation of the Newton s law of univesal gavitation. The deivation is elativisti beause we used the total elativisti enegy and of the attating bodies. At the sae tie the deivation is quantu ehanial beause we intodued the lank foe ( ). Howeve the law we obtained is a lassial law. Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino. 6
Taking into aount that the sale law desibes seveal known laws of physis as I have shown both on pevious papes [,,,, 5] and on this pape, we an onside the sale law as a oe univesal law than the speifi laws it desibes. REERENCE [] R. A. ino, ale atos and the ale iniple, vixa: 05.070, (0). [] R. A. ino, The peial Theoy of Relativity and the ale iniple, vixa: 06.0, (0). [] R. A. ino, The hwazhild Radius and the ale iniple, vixa: 06.06, (0). [] R. A. ino, The ine tutue Constant and the ale iniple, vixa: 06.069, (0). [5] R. A. ino, The Boh ostulate, the De Boglie Condition and the ale iniple, vixa:, (0). Newton s Law of Univesal avitation and the ale iniple v. Copyight 0 Rodolfo A. ino. 7